Empty Relations: Domain, Range & Functionality

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Discussion Overview

The discussion revolves around the concept of empty relations in the context of set theory and functions. Participants explore whether an empty relation can be classified as a function, the implications of having an empty set as a domain or range, and the definitions surrounding these concepts.

Discussion Character

  • Debate/contested
  • Conceptual clarification

Main Points Raised

  • Some participants assert that an empty relation is a subset of the Cartesian product AxA and thus qualifies as a relation.
  • It is proposed that an empty relation can be a function since the definition allows for the possibility of having no pairs.
  • One participant challenges the definition of a function, suggesting it requires exactly one pair for each element in the domain, rather than at most one.
  • There is a call for clarification on the term "range," indicating that it may have multiple interpretations in this context.
  • Another viewpoint emphasizes the lack of a universal definition for "function" or "relation," suggesting that definitions can vary and must be adhered to once chosen.
  • One participant expresses a belief that functions should not be considered a type of relation, highlighting grammatical differences in their definitions.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the definitions and implications of empty relations and functions. Multiple competing views remain regarding the nature of these concepts and their interrelations.

Contextual Notes

Limitations include the ambiguity in definitions of "function" and "relation," as well as the varying interpretations of "range." These factors contribute to the unresolved nature of the discussion.

luizgguidi
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What is an empty relation?
Can an empty relation be a function?
Is an empty relation one with the empty set as its domain or as its range or both?

THanks
 
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Look at the definitions! (How many times have I said that?)

A "relation" on a set A is a subset of the Cartesian product AxA. Since the empty set is a subset of any set, yes, the empty set is a subset of AxA and so is a relation.

A "function", F, is a relation on AxA such that "if x is in A, there is not more than one pair (x,y) with first member x". Because that says "not more than" it includes none. Yes, that is a function.

Any relation that has the empty set as its domain MUST have the empty set as its range and also is the empty relation.
 
HallsofIvy said:
A "function", F, is a relation on AxA such that "if x is in A, there is not more than one pair (x,y) with first member x".
Actually, it's "there is exactly one pair".

And we should ask the original poster what precisely he means by "range" -- there are at least two distinct ways of using it here.
 
luizgguidi said:
What is an empty relation?
Can an empty relation be a function?
Is an empty relation one with the empty set as its domain or as its range or both?

It's not really useful to worry about trivial points like this. It's all a matter of definition, and there is no universal definition for "function" or "relation". You can use any definition you want, but once you choose one, you have to stick with it.

In my mind, at least, I wouldn't even consider functions to be a kind of relation anyway. They are often defined in terms of relations, but they are very different grammatically. If R is a relation, "x R y" is a sentence. It can be either true or false. If f is a function, "f(x)" is just a noun. It has no statement behind it.
 

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